Complex Hadamard Diagonalisable Graphs

TL;DR

This paper extends the concept of Hadamard diagonalisable graphs to complex Hadamard matrices, exploring their properties, constructions, and applications in quantum state transfer.

Contribution

It introduces complex Hadamard diagonalisable graphs, analyzes their properties, and provides new constructions and conditions for quantum state transfer.

Findings

Laplacian eigenvalues are even integers for these graphs

Vertex sets often form equitable partitions

Examples include Cayley graphs and NEPS

Abstract

In light of recent interest in Hadamard diagonalisable graphs (graphs whose Laplacian matrix is diagonalisable by a Hadamard matrix), we generalise this notion from real to complex Hadamard matrices. We give some basic properties and methods of constructing such graphs. We show that a large class of complex Hadamard diagonalisable graphs have vertex sets forming an equitable partition, and that the Laplacian eigenvalues must be even integers. We provide a number of examples and constructions of complex Hadamard diagonalisable graphs, including two special classes of graphs: the Cayley graphs over $\mathbb{Z}_r^d$, and the non--complete extended $p$--sum (NEPS). We discuss necessary and sufficient conditions for $(\alpha, \beta)$--Laplacian fractional revival and perfect state transfer on continuous--time quantum walks described by complex Hadamard diagonalisable graphs and provide examples of such quantum state transfer.